Euclidean preferences in the plane under \(\varvec{\ell _1},\) \(\varvec{\ell _2}\) and \(\varvec{\ell _\infty }\) norms

We present various results about Euclidean preferences in the plane under \(\ell _1,\) ${\ell }_1,$ \(\ell _2\) ${\ell }_2$ and \(\ell _{\infty }\) ${\ell }_{\infty }$ norms. When there are four candidates, we show that the maximum size (in terms of the number of pairwise distinct preferences) of Euclidean preference profiles in \({\mathbb {R}}^2\) ${\mathbb{R}}^2$ under norm \(\ell _1\) ${\ell }_1$ or \(\ell _{\infty }\) ${\ell }_{\infty }$ is 19. Whatever the number of candidates, we prove that at most four distinct candidates can be ranked in the last position of a two-dimensional Euclidean preference profile under norm \(\ell _1\) ${\ell }_1$ or \(\ell _\infty ,\) ${\ell }_{\infty },$ which generalizes the case of one-dimensional Euclidean preferences (for which it is well known that at most two candidates can be ranked last). We generalize this result to \(2^d\) $2^d$ (resp. 2d) for \(\ell _1\) ${\ell }_1$ (resp. \(\ell _\infty \) ${\ell }_{\infty }$ ) for d-dimensional Euclidean preferences. We also establish that the maximum size of a two-dimensional Euclidean preference profile on m candidates under norm \(\ell _1\) ${\ell }_1$ is in \(\varTheta (m^4),\) $\Theta (m^4),$ which is the same order of magnitude as the known maximum size under norm \(\ell _2.\) ${\ell }_2.$ Finally, we provide a new proof that two-dimensional Euclidean preference profiles under norm \(\ell _2\) ${\ell }_2$ for four candidates can be characterized by three inclusion-maximal two-dimensional Euclidean profiles. This proof is a simpler alternative to that proposed by Kamiya et al. (Adv Appl Math 47(2):379–400, 2011).